There are a lot of things to take into account here.
For Legolas: His strengths would obviously include that he is thousands of years old, and has a lot of training. He is extremely fast, accurate, and agile, and could probably swing 3 times for every single swing of Gimli. However, there are also weaknesses to take into account. As the extended edition of battle of the 5 armies demonstrated, the elves are arrogant, cocky even. I would highly expect that Legolas would only be used to fighting lowly creatures like orcs, not a true threat like Gimli and would underestimate him. Also, dwarven armor is Legendary
, they are without contest the BEST armor smiths in Middle-Earth, so I have serious doubts that any arrows fired by Legolas could even pierce Gimli's armor, it would have to come down to close combat, which is just where the dwarf wants to be. Gimli also isn't just going to stand there and take an arrow to the face, so I discount that possibility.
As for Gimli, the only thing that needs to be said separately from above is that he is obviously the more durable and stubborn. Legolas winning the drinking game was nonsense, alcohol effects Elves, as evidenced by The Hobbit, both book and film, so I completely discount that argument. It was just a funny moment in the film, not canon. Gimli's sheer endurance, combined with his armor, means that he could probably take many hits from Legolas before even being slowed down, whereas a single hit from his battleaxe or even his thrown axe would severely injure Legolas.
Finally, a shared advantage is that both are heroes, and therefore have what I call "Heroic Luck". Basically, attacks that would be a sure-fire hit against an orc somehow seem to miss them, or glance off. So Legolas could be presumed to be half as likely to hit with his bow, same goes for Gimli with throwing axes.
Ultimately, I think that Gimli simply has more advantages. Better armor and endurance, a more dangerous close combat melee weapon, and finally he would be more likely to actually WANT to kill the Elf, and wouldn't underestimate him. I give this match to Gimli.
Here is my proposal: Gandalf the White versus the Witch King. Not the ridiculous nonsense in the film where his staff just blows up, Gandalf is a freaking Maiar, a lesser GOD, on par with Sauron himself. The Witch King is just an old man who knows some magic, and his greatest power was that of fear, which obviously does not affect Gandalf. Take it away